Problem 10 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{-24 a^{3} b^{3}}{39 a^{5} b^{2}}$$

Step-by-Step Solution

Answer

\(\frac{-8b}{13a^{2}}\)

Step 1: Rewrite the Fraction

Begin by writing down the fraction as it is given: \[\frac{-24 a^{3} b^{3}}{39 a^{5} b^{2}}\]

Step 2: Factor Out Common Terms

Identify the common terms in both the numerator and the denominator. Here, both contain terms with \(a\) and \(b\).

Step 3: Cancel Common Factors for \(a\)

Reduce the powers of \(a\) by subtracting the exponent in the denominator from the exponent in the numerator: \[\frac{-24}{39} \times \frac{a^{3}}{a^{5}} = \frac{-24}{39} \times a^{3-5} = \frac{-24}{39} \times \frac{1}{a^{2}}\]

Step 4: Cancel Common Factors for \(b\)

Do the same for \(b\) by subtracting the exponents: \[\frac{-24}{39} \times \frac{b^{3}}{b^{2}} = \frac{-24}{39} \times b^{3-2} = \frac{-24}{39} \times b^{1}\]

Step 5: Simplify the Coefficients

Simplify the fraction with the numerical coefficients: \[\frac{-24}{39}\] Find the greatest common factor (GCF) of 24 and 39, which is 3. Divide both numbers by their GCF:\[\frac{-24 ÷ 3}{39 ÷ 3} = \frac{-8}{13}\]

Step 6: Write the Simplified Fraction

Combine the simplified numerical coefficient with the simplified variable terms:\[\frac{-8}{13} \times \frac{b}{a^{2}} = \frac{-8b}{13a^{2}}\] This is the simplified form of the algebraic fraction.

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complicated algebraic fractions into their simplest form. This process helps in better understanding and solving algebraic problems by making expressions more manageable. When simplifying expressions,

• identify terms that can be combined or reduced,
• make sure that any common factors in numerators and denominators are canceled,
• and simplify any coefficients wherever possible.

For algebraic fractions, this might involve factoring or finding and eliminating common factors between terms.

Factoring

Factoring refers to finding components that multiply together to form an expression. In simpler cases, it's like reverse distributing the multiplication. For the given problem, notice the terms in both the numerator and the denominator: - These terms include variables with exponents and sometimes constants. - Factoring is useful because it helps in determining what can be canceled out. Consider the case of extracting any simple expressions or integers that repeat in both the top and the bottom parts of the fraction.

Exponents

Exponents are an essential part of algebra that signal how many times a number or variable is used as a factor. When simplifying algebraic fractions with exponents, it's crucial to understand the rules:

• When dividing with like bases, you subtract the exponents (e.g., \(a^{m} / a^{n} = a^{m-n}\)).
• This subtraction of exponents allows us to cancel out similar base components in a fraction, reducing complexity.

In our exercise, subtracting the exponents of \(a\) and \(b\) helped to simplify the expression substantially.

Common Factors

Common factors appear when there are identical terms in both the numerator and the denominator. These terms can be algebraic (variables with exponents) or numeric (coefficients). The process of identifying common factors: - Provides a method to simplify fractions by removing equal parts from both sides of the fraction. - Which helps in minimizing the expression. In the case of the given fraction, we saw the numerators and denominators having common exponents and coefficients. By factoring out and canceling these common factors, we end up with a simplified algebraic fraction.